1. GEOMAGNETISM: a dynamo at the centre of the Earth Lecture 1 How the dynamo is powered Lecture 2 How the dynamo works Lecture 3 Interpreting the observations Lecture 4 Thermal core-mantle interactions

2. Lecture 1 How the dynamo is powered Gubbins, D., D. Alfe, G. Masters, D Price & M.J. Gillan “Can the Earth’s dynamo run on heat alone?” “Gross thermodynamics of 2-component core convection” - both under review for Geophysical Journal International

3. ENERGY LOST THROUGH ELECTRICAL RESISTANCE Magnetic field decays in 15,000 years Energy loss is 10 11 - 10 12 W

4. THE MODEL Core cooling drives convection Perhaps some radioactive heating Inner core freezes -> more latent heat… …and releases light material that drives convection through… Release of gravitational energy

5. inner core O H S Si K 40 Fe Mantle latent heat

6. THE BASIC STATE Pressure is nearly hydrostatic: Convective velocity >> diffusion… …means core is well mixed …including entropy Temperature is adiabatic

7. GRUNEISSEN’S PARAMETER Thermodynamic definition: Hydrostatic pressure: Seismic parameter:

8. Temperature in the core is found by integrating up from the inner core boundary, where T is the known melting temperature Time evolution of the (logarithm) of temperature is then the same everywhere:

9. INNER CORE FREEZING

10. THE FIRST TWIST... Conservation of energy does not equate the energy required with the heat lost by the magnetic field, in fact it does not involve the magnetic field at all!

11. ENERGY FLOW CHART dynamo electrical heating expansion buoyancy conduction + convection

12. ENTROPY BALANCE Dissipation gives entropy gains: thermal conduction electrical conduction Offset by entropy losses if T in >T out

13. BACKUS’ IDEAL DYNAMO “Efficiency” : Can be greater than unity. This is because the output of the heat engine, the electrical heating, is used again in powering the convection. A Carnot engine driving a disk dynamo achieves the ideal bound

14. THE SECOND TWIST... Cooling and contraction releases a significant amount of Earth’s gravitational energy Freezing also releases gravitational energy Is this available to the dynamo? Some think so But only about 5% is available

15. GRAVITATIONAL ENERGY Is calculated from the work done in assembling all the mass from infinity The gravitational force is conservative, so we can do this however we like Assemble the mass of the Earth slowly, maintaining hydrostatic pressure Then all of the gravitational energy goes into compaction, except for…. …a small amount caused by pressure heating

16. PRESSURE HEATING Drop temperature for change in volume From the Maxwell relation Heat released Divide by specific heat released:

17. PRESSURE EFFECT ON FREEZING The change in volume on freezing also releases gravitational energy The change in volume on freezing is related to the latent heat (L) through the Clausius-Clapeyron equation Again the only part of this gravitational energy that is available to drive convection is a small amount of pressure heating

18. The increase in melting temperature caused by the higher causes the inner core to grow a little more The latent heat released is identically equal to the gravitational energy change, because of the Clausius- Clapeyron equation

19. SUMMARY - HEAT ONLY Entropy balance: choose LHS and find cooling rate and radioactive heating h Energy balance: find heat flux from cooling rate and radioactive heating h

21. ADIABATIC GRADIENTS

22. HEAT BUDGET Earth’s heat budget: Crustal radioactivity9 TW mostly lower crust mantle radioactivity 25 TWchondritic composition core radioactivity0 TWiron meteorites, chemistry cooling 10 TW includes core, mantle TOTAL 44 TW Surface heat flux Cooling rate: 36 K/GyrFrom core 3 TW

23. RESULTS FOR THERMAL CONVECTION MODELdTc/dt dri/dtICage Q L Q S Q K/Gyr km/GyrMaTWTWTW GAMP02214 141428810.210.921.6 LPL97234 15502638.414.223.0 NOIC565 0.028.828.8 Comparison between 3 models of Gubbins et al 2002; LaBrosse et al 1997 (modified); and a model with no inner core (L=0)

24. COMPOSITIONAL CONVECTION Light material released at the inner core boundary on freezing rises to stir the core Energy source is Earth’s gravitational energy This changes as light material rises, heavy iron sinks Compositional convection stirs the core directly, there is no thermal efficiency factor

25. inner core O H S Si K 40 Fe Mantle latent heat

27. THE STORY SO FAR... Thermal convection cannot drive the dynamo because too much heat is needed This means we have no means of generating a magnetic field before the inner core formed, the inner core must be as old as the magnetic field Compositional convection can help drive the dynamo The solid inner core can include 8% S or Si to explain the density. When this mixture freezes, it all freezes. A liquid Fe+8%S+8% O can explain the density of the liquid outer core When Fe+8%O mixture freezes, the O is left in the liquid This provides the source of buoyancy for compositional convection

28. NEXT... We see if compositional plus thermal convection can drive the dynamo We estimate the cooling rates and radioactive heating needed by balancing the entropy Then we use the cooling rate and radioactive heating to calculate the heat flux across the core- mantle boundary and the inner core age.

29. CORE COMPOSITION OF PRICE, ALFE & GILLAN (2001)

30. DENSITY REDUCTIONS FROM PURE IRON AT ICB PRESSURE AND TEMPERATURE  Solid iron13.16 8% S/Si12.763.0%0.40 Melting12.521.8%0.24 8%O12.172.8%0.37 Ideal solutions theory predicts densities well, but not diffusion constants or free energies

31. DISSIPATION ENTROPY Thermal conduction 200-500 MW/K Ohmic heating 50-500 MW/K Molecular diffusion 1 MW/K Round up 1000 MW/K

32. And find the heat flux from cooling rate and radioactive heating Find the cooling rate and radioactive heating from the entropy balance FINAL EQUATIONS

33. THE MODELS E = 1000 MW/K “rounding up” E= 546 MW/K, heat conducted down by compositional convection E = 262 MW/KDynamo fails Repeat with enough radioactive heating to make the inner core last 3.5 Gyr

34. RESULTS FOR COMPOSITIONAL CONVECTION

35. CONCLUSIONS Compositional convection only doubles the efficiency of the dynamo With present estimates and no radioactivity in the core, the age of the inner core is less than 1Ga The simplest way to alter this result is to increase the seismological estimate of the density jump at the inner core boundary At present it seems impossible to drive the dynamo without an inner core